Path-metrics, pruning, and generalization

Analyzing the behavior of ReLU neural networks often hinges on understanding the relationships between their parameters and the functions they implement. This paper proves a new bound on function distances in terms of the so-called path-metrics of the parameters. Since this bound is intrinsically invariant with respect to the rescaling symmetries of the networks, it sharpens previously known bounds. It is also, to the best of our knowledge, the first bound of its kind that is broadly applicable to modern networks such as ResNets, VGGs, U-nets, and many more. In contexts such as network pruning and quantization, the proposed path-metrics can be efficiently computed using only two forward passes. Besides its intrinsic theoretical interest, the bound yields not only novel theoretical generalization bounds, but also a promising proof of concept for rescaling-invariant pruning.

Keep the Momentum: Conservation Laws beyond Euclidean Gradient Flows

Conservation laws are well-established in the context of Euclidean gradient flow dynamics, notably for linear or ReLU neural network training. Yet, their existence and principles for non-Euclidean geometries and momentum-based dynamics remain largely unknown. In this paper, we characterize "all" conservation laws in this general setting. In stark contrast to the case of gradient flows, we prove that the conservation laws for momentum-based dynamics exhibit temporal dependence. Additionally, we often observe a "conservation loss" when transitioning from gradient flow to momentum dynamics. Specifically, for linear networks, our framework allows us to identify all momentum conservation laws, which are less numerous than in the gradient flow case except in sufficiently over-parameterized regimes. With ReLU networks, no conservation law remains. This phenomenon also manifests in non-Euclidean metrics, used e.g. for Nonnegative Matrix Factorization (NMF): all conservation laws can be determined in the gradient flow context, yet none persists in the momentum case.

Sketch and shift: a robust decoder for compressive clustering

Compressive learning is an emerging approach to drastically reduce the memory footprint of large-scale learning, by first summarizing a large dataset into a low-dimensional sketch vector, and then decoding from this sketch the latent information needed for learning. In light of recent progress on information preservation guarantees for sketches based on random features, a major objective is to design easy-to-tune algorithms (called decoders) to robustly and efficiently extract this information. To address the underlying non-convex optimization problems, various heuristics have been proposed. In the case of compressive clustering, the standard heuristic is CL-OMPR, a variant of sliding Frank-Wolfe. Yet, CL-OMPR is hard to tune, and the examination of its robustness was overlooked. In this work, we undertake a scrutinized examination of CL-OMPR to circumvent its limitations. In particular, we show how this algorithm can fail to recover the clusters even in advantageous scenarios. To gain insight, we show how the deficiencies of this algorithm can be attributed to optimization difficulties related to the structure of a correlation function appearing at core steps of the algorithm. To address these limitations, we propose an alternative decoder offering substantial improvements over CL-OMPR. Its design is notably inspired from the mean shift algorithm, a classic approach to detect the local maxima of kernel density estimators. The proposed algorithm can extract clustering information from a sketch of the MNIST dataset that is 10 times smaller than previously.

Revisiting RIP guarantees for sketching operators on mixture models

In the context of sketching for compressive mixture modeling, we revisit existing proofs of the Restricted Isometry Property of sketching operators with respect to certain mixtures models. After examining the shortcomings of existing guarantees, we propose an alternative analysis that circumvents the need to assume importance sampling when drawing random Fourier features to build random sketching operators. Our analysis is based on new deterministic bounds on the restricted isometry constant that depend solely on the set of frequencies used to define the sketching operator; then we leverage these bounds to establish concentration inequalities for random sketching operators that lead to the desired RIP guarantees. Our analysis also opens the door to theoretical guarantees for structured sketching with frequencies associated to fast random linear operators.

Compressive Recovery of Sparse Precision Matrices

We consider the problem of learning a graph modeling the statistical relations of the $d$ variables of a dataset with $n$ samples $X \in \mathbb{R}^{n \times d}$. Standard approaches amount to searching for a precision matrix $\Theta$ representative of a Gaussian graphical model that adequately explains the data. However, most maximum likelihood-based estimators usually require storing the $d^{2}$ values of the empirical covariance matrix, which can become prohibitive in a high-dimensional setting. In this work, we adopt a compressive viewpoint and aim to estimate a sparse $\Theta$ from a sketch of the data, i.e. a low-dimensional vector of size $m \ll d^{2}$ carefully designed from $X$ using nonlinear random features. Under certain assumptions on the spectrum of $\Theta$ (or its condition number), we show that it is possible to estimate it from a sketch of size $m=\Omega((d+2k)\log(d))$ where $k$ is the maximal number of edges of the underlying graph. These information-theoretic guarantees are inspired by compressed sensing theory and involve restricted isometry properties and instance optimal decoders. We investigate the possibility of achieving practical recovery with an iterative algorithm based on the graphical lasso, viewed as a specific denoiser. We compare our approach and graphical lasso on synthetic datasets, demonstrating its favorable performance even when the dataset is compressed.

A path-norm toolkit for modern networks: consequences, promises and challenges

This work introduces the first toolkit around path-norms that is fully able to encompass general DAG ReLU networks with biases, skip connections and any operation based on the extraction of order statistics: max pooling, GroupSort etc. This toolkit notably allows us to establish generalization bounds for modern neural networks that are not only the most widely applicable path-norm based ones, but also recover or beat the sharpest known bounds of this type. These extended path-norms further enjoy the usual benefits of path-norms: ease of computation, invariance under the symmetries of the network, and improved sharpness on feedforward networks compared to the product of operators' norms, another complexity measure most commonly used. The versatility of the toolkit and its ease of implementation allow us to challenge the concrete promises of path-norm-based generalization bounds, by numerically evaluating the sharpest known bounds for ResNets on ImageNet.

Abide by the Law and Follow the Flow: Conservation Laws for Gradient Flows

Understanding the geometric properties of gradient descent dynamics is a key ingredient in deciphering the recent success of very large machine learning models. A striking observation is that trained over-parameterized models retain some properties of the optimization initialization. This "implicit bias" is believed to be responsible for some favorable properties of the trained models and could explain their good generalization properties. The purpose of this article is threefold. First, we rigorously expose the definition and basic properties of "conservation laws", which are maximal sets of independent quantities conserved during gradient flows of a given model (e.g. of a ReLU network with a given architecture) with any training data and any loss. Then we explain how to find the exact number of these quantities by performing finite-dimensional algebraic manipulations on the Lie algebra generated by the Jacobian of the model. Finally, we provide algorithms (implemented in SageMath) to: a) compute a family of polynomial laws; b) compute the number of (not necessarily polynomial) conservation laws. We provide showcase examples that we fully work out theoretically. Besides, applying the two algorithms confirms for a number of ReLU network architectures that all known laws are recovered by the algorithm, and that there are no other laws. Such computational tools pave the way to understanding desirable properties of optimization initialization in large machine learning models.

About the Cost of Global Privacy in Density Estimation

We study non-parametric density estimation for densities in Lipschitz and Sobolev spaces, and under global privacy. In particular, we investigate regimes where the privacy budget is not supposed to be constant. We consider the classical definition of global differential privacy, but also the more recent notion of global concentrated differential privacy. We recover the result of Barber \& Duchi (2014) stating that histogram estimators are optimal against Lipschitz distributions for the L2 risk, and under regular differential privacy, and we extend it to other norms and notions of privacy. Then, we investigate higher degrees of smoothness, drawing two conclusions: First, and contrary to what happens with constant privacy budget (Wasserman \& Zhou, 2010), there are regimes where imposing privacy degrades the regular minimax risk of estimation on Sobolev densities. Second, so-called projection estimators are near-optimal against the same classes of densities in this new setup with pure differential privacy, but contrary to the constant privacy budget case, it comes at the cost of relaxation. With zero concentrated differential privacy, there is no need for relaxation, and we prove that the estimation is optimal.

Does a sparse ReLU network training problem always admit an optimum?

Given a training set, a loss function, and a neural network architecture, it is often taken for granted that optimal network parameters exist, and a common practice is to apply available optimization algorithms to search for them. In this work, we show that the existence of an optimal solution is not always guaranteed, especially in the context of {\em sparse} ReLU neural networks. In particular, we first show that optimization problems involving deep networks with certain sparsity patterns do not always have optimal parameters, and that optimization algorithms may then diverge. Via a new topological relation between sparse ReLU neural networks and their linear counterparts, we derive -- using existing tools from real algebraic geometry -- an algorithm to verify that a given sparsity pattern suffers from this issue. Then, the existence of a global optimum is proved for every concrete optimization problem involving a shallow sparse ReLU neural network of output dimension one. Overall, the analysis is based on the investigation of two topological properties of the space of functions implementable as sparse ReLU neural networks: a best approximation property, and a closedness property, both in the uniform norm. This is studied both for (finite) domains corresponding to practical training on finite training sets, and for more general domains such as the unit cube. This allows us to provide conditions for the guaranteed existence of an optimum given a sparsity pattern. The results apply not only to several sparsity patterns proposed in recent works on network pruning/sparsification, but also to classical dense neural networks, including architectures not covered by existing results.

Private Statistical Estimation of Many Quantiles

This work studies the estimation of many statistical quantiles under differential privacy. More precisely, given a distribution and access to i.i.d. samples from it, we study the estimation of the inverse of its cumulative distribution function (the quantile function) at specific points. For instance, this task is of key importance in private data generation. We present two different approaches. The first one consists in privately estimating the empirical quantiles of the samples and using this result as an estimator of the quantiles of the distribution. In particular, we study the statistical properties of the recently published algorithm introduced by Kaplan et al. 2022 that privately estimates the quantiles recursively. The second approach is to use techniques of density estimation in order to uniformly estimate the quantile function on an interval. In particular, we show that there is a tradeoff between the two methods. When we want to estimate many quantiles, it is better to estimate the density rather than estimating the quantile function at specific points.